Multidimensional systems

The Heisler charts discussed above may be used to obtain the temperature distribution in the infinite plate of thickness 2L, in the long cylinder, or in the sphere. When a wall whose height and depth dimensions are not large compared with the thickness or a cylinder whose length is not large compared with its diameter is encountered, additional space coordinates are necessary to specify the temperature, the above charts no longer apply, and we are forced to seek another method of solution. Fortunately, it is possible to combine the solutions 

It is clear that the infinite rectangular bar in Fig. 4-17 can be formed from two infinite plates of thickness 2L, and 2L2, respectively. The differential equation governing this situation would be 

It can be shown that the dimensionless temperature distribution may be expressed as a product of the solutions for two plate problems of thickness 2Lt and 2Li, respectively  

For two infinite plates the respective differential equations would be 

We shall now show that the product solution to Eq. (4-17) can be formed from a simple product of the functions (T, T2), that is, 

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The calculation of the time evolution operator in multidimensional systems is a fomiidable task and some results will be discussed in this section. An alternative approach is the calculation of semi-classical dynamics as demonstrated, among others, by Heller [86, 87 and 88], Marcus [89, 90], Taylor [91, 92], Metiu [93, 94] and coworkers (see also [83] as well as the review by Miller [95] for more general aspects of semiclassical dynamics). This method basically consists of replacing the 5-fimction distribution in the true classical calculation by a Gaussian distribution in coordinate space. It allows for a simulation of the vibrational... 

Kay K G 1994 Semiclassical propagation for multidimensional systems by an initial value method J. Chem. Phys. 101 2250... 

In this appendix we shall show how the quasienergy ideas developed in section 5 can be applied to the problem of nonadiabatic tunneling. We use the Imf approach of section 3.3 for the multidimensional system with Hamiltonian... 

Multidimensional LC separation has been defined as a teehnique whieh is mainly eharaeterized by two distinet eriteria, as follows (1). The first eriterion for a multidimensional system is that sample eomponents must be displaeed by two or more separation teehniques involving orthogonal separation meehanisms (2), while the seeond eriterion is that eomponents that are separated by any single separation dimension must not be reeombined in any further separation dimension. 

More generally the peak capacity for a multidimensional system can be expressed by the following ... 

T. Greibrokk, Applications of supercritical fluid exti action in multidimensional systems , J. Chromatogr. 703 523-536 (1995). 

ADVANTAGES OF AND CHALLENGES FOR UNIFIED CHROMATOGRAPHY TECHNIQUES IN MULTIDIMENSIONAL SYSTEMS... 

The point of all this is simply that we must not use the apparent plate height or the apparent plate number as performance criteria in the unified chromatography techniques on the justification that they already work well for LC and that they work well for GC when a pressure correction is applied. A considerable expansion of theory and an effective means for evaluating equations (7.4) or (7.5) are required first. Likewise, as we consider multidimensional chromatography involving techniques existing between the extremes of LC and GC, we must not build judgments of the multidimensional system on unsound measures of the individual techniques involved. 

MD-PC is highly important in its own right, because this is the only real multidimensional separation method in which all compounds can be passed to a next dimension. It therefore serves as the reference system (7) against which all other multidimensional systems can be compared. 

Electrodriven separation techniques are destined to be included in many future multidimensional systems, as CE is increasingly accepted in the analytical laboratory. The combination of LC and CE should become easier as vendors work towards providing enhanced microscale pumps, injectors, and detectors (18). Detection is often a problem in capillary techniques due to the short path length that is inherent in the capillary. The work by Jorgenson s group mainly involved fluorescence detection to overcome this limit in the sensitivity of detection, although UV-VIS would be less restrictive in the types of analytes detected. Increasingly sensitive detectors of many types will make the use of all kinds of capillary electrophoretic techniques more popular. 

Another way to improve the analysis of complex matrices can be the combination of a multidimensional system with information-rich spectral detection (31). The analysis of eucalyptus and cascarilla bark essential oils has been carried out with an MDGC instrument, coupling a fast second chromatograph with a matrix isolation infrared spectrometer. Eluents from the first column were heart-cut and transferred to a cryogenically cooled trap. The trap is then heated to re-inject the components into an analytical column of different selectivity for separation and subsequent detection. The problem of the mismatch between the speed of fast separation and the... 

A multidimensional system using capillary SEC-GC-MS was used for the rapid identification of various polymer additives, including antioxidants, plasticizers, lubricants, flame retardants, waxes and UV stabilizers (12). This technique could be used for additives having broad functionalities and wide volatility ranges. The determination of the additives in polymers was carried out without performing any extensive manual sample pretreatment. In the first step, microcolumn SEC excludes the polymer matrix from the smaller-molecular-size additives. There is a minimal introduction of the polymer into the capillary GC column. Optimization of the pore sizes of the SEC packings was used to enhance the resolution between the polymer and its additives, and smaller pore sizes could be used to exclude more of the polymer... 

The use of multidimensional chromatography in environmental analysis has been reviewed in the literature (1-6). Of the multidimensional systems described in previous chapters, GC-GC liquid chromatography LC-LC and LC-GC, whose applications to environmental analysis will be detailed in this chapter, are the ones most often used in environmental analysis. 

Other multidimensional systems, such as supercritical fluid chromatography (SFC-GC or LC-SFC), will not be described here because, although some applications to environmental analysis have been described (4, 7-9), they have not been very widely used in this field. 

Multidimensional LC-LC, using two high-resolution eolumns with orthogonal separation meehanisms, has only a few applieations in environmental analysis. The limitations that sueh a multidimensional system has with regard to seleetivity must... 

In considering the equivalent of eq. (2.8) for multidimensional systems we will start by defining the relevant reaction coordinate, X, and the probability, P(X), that the system will be at different points, along X. The reaction coordinate can be taken rather arbitrarily as any well-defined parameter fe.g., X = (r23 - r12) in Fig. 1.7]. Once X is selected we can obtain P(X) by dividing the coordinate space into subsets according to the specific value of X and evaluating the one-dimensional function. 

In the same way as described above, we can formulate the multidimensional theory without relying on the complex-valued Lagrange manifold that constitutes one of the main obstacles of the conventional multidimensional WKB theory [62,63,77,78]. Another crucial point is that the theory should not depend on any local coordinates, which gives a cumbersome problem in practical applications. Below, a general formulation is described, which is free from these difficuluties and applicable to vertually any multidimensional systems [30]. 

In the case of longitudinal excitation Eqs. (69) and (70) hold true in a general multidimensional system, as explained later. 

Thus, If two identical coluwis with a peak capacity of 25 are coupled in series, then the resultant peak capacity would be about 35, conpared to a value of 625 if the same columns were used in the multidimensional mode. In many Instances formidable technical problems must be solved to take full advantage of the potential of multidimensional systems (section 8.7). 

The objective of combined analytical separations is to obtain nonredundant information from independent systems. The success of all multidimensional methods in chromatography is dependent on the creation of complementary separation mechanisms, applied in a sequential manner, to enhance the separation capacity of the system. For techniques to be complementary to each other, the acquired data should be orthogonal. A multidimensional system is commonly defined as a system in which ... 

Cortes et al. [18] have quantitatively determined polymer additives in a polycarbonate homopolymer and an ABS terpolymer. In that case, a multidimensional system consisting of a microcolumn SEC was coupled on-line to either capillary GC or a conventional LC system. The results show losses of certain additives when using the conventional precipitation approach. An at-column GC procedure has been developed for rapid determination (27 min) of high-MW additives (ca. 1200Da) at low concentrations (lOOppm) in 500- xL SEC fractions in DCM for on-line quality control (RSD of 2-7%) [36], Also, SEC-NPLC has been used for the analysis of additives in dissolution of polymeric... 

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